Abstract
We study a rational decision maker who obeys social norms. In our setup norms prescribe choices in some decision problems. The decision maker obeys norms in situations to which they apply and otherwise maximizes her preference relation. We characterize the class of choice functions that can be explained by this decision procedure, relate this procedure to other decision procedures in the literature, and engage in welfare considerations.
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Notes
Abusing notation, we will write \(C\left( x_{1},\dots ,x_{k}\right) \) instead of \(C\left( \left\{ x_{1},\dots ,x_{k}\right\} \right) \) for \(x_{j}\in X,\,j=1,\dots ,k\).
That is, \(\succ \) is asymmetric, transitive, and satisfies the following completeness property: \(\forall x,y\in X,x\ne y\) it holds \(x\succ y\) or \(y\succ x\).
To see the details, assume that \(X=\left\{ a,b,c\right\} \) and \(C\) is induced by the sequence of rationales \(\left( P_{i}\right) _{i\in \mathbb {N}} \). W.l.o.g. we can assume \(P_{1}\ne \varnothing \). \(C\left( a,b,c\right) =b\) implies \(\left( a,b\right) \notin P_{1}\) and \(\left( c,b\right) \notin P_{1}\). \(C\left( a,b\right) =a\) implies \(\left( b,a\right) \notin P_{1}\) and \(C\left( a,c\right) =a\) implies \(\left( c,a\right) \notin P_{1}\). Hence \(\left( a,c\right) \in P_{1}\) or \(\left( b,c\right) \in P_{1}\). Either way we have \(\gamma _{P_{1}}\left( a,b,c\right) =\gamma _{P_{1}}\left( a,b\right) \), contradicting \(C\left( a,b,c\right) \ne C\left( a,b\right) \).
Note that Cherepanov et al. (2013) actually derive In from some deeper structure modelling the use of rationalizations in decision making. However for our purposes this derivation is not of interest.
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Thanks to Bea, who suggested the term ‘obedience’, and two referees for good advice.
Appendix
Appendix
Proposition 5
Consider the following choice function: \(C_{1}\left( a,b,c\right) =a\), \(C_{1}\left( a,b\right) =a\), and \(C_{1}\left( b,c\right) =C_{1}\left( a,c\right) =c\). \(C_{1}\) satisfies OD but not AC.
Consider \(C_{2}\left( a,b\right) =C_{2}\left( a,d\right) =a\), \(C_{2}\left( a,c\right) =C_{2}\left( c,d\right) =c\), \(C_{2}\left( b,c\right) =C_{2}\left( b,d\right) =b\). Letting \(C_{2}\left( a,b,c\right) =C_{2}\left( b,c,d\right) =b\), \(C_{2}\left( a,b,d\right) \!=\!C_{2}\left( a,b,c,d\right) =a\), and \(C_{2}\left( a,c,d\right) =c\), we observe that \(C_{2}\) satisfies AC, but fails OD with respect to \(\left\{ a,b,c,d\right\} ,\left\{ a,b,c\right\} \), and \(\left\{ a,b\right\} \).
Let \(C_{3}\left( a,b,c\right) =C_{3}\left( a,b\right) =a\), \(C_{3}\left( b,c\right) =b\), and \(C_{3}\left( a,c\right) =c\) meets OD but defys NBC. On the other hand, \(C_{4}\left( S\right) =C_{2}\left( S\right) \) for all \(S\in \mathbb {X}\backslash \left\{ \left\{ a,c\right\} \right\} \) and \(C_{4}\left( a,c\right) =a\) satisfies NBC, but violates OD.
Let \(C_{5}\left( a,b,c,d\right) =C_{5}\left( a,b,d\right) =C_{5}\left( a,c,d\right) =C_{5}\left( a,d\right) =a\), \(C_{5}\left( a,b,c\right) =C_{5}\left( b,c,d\right) =b\), \(C_{5}\left( a,b\right) =C_{5}\left( b,d\right) =b\), and \(C_{5}\left( b,c\right) =C_{5}\left( a,c\right) =C_{5}\left( c,d\right) =c\). \(C_{5}\) satisfies WW, but violates OD with respect to \(\left\{ a,b,c,d\right\} ,\left\{ a,b,c\right\} \), and \(\left\{ b,c\right\} \).
Section 3.1
Consider the following semiorders: \(P_{1}=\left\{ \left( d,b\right) ,\left( d,c\right) \right\} \), \(P_{2}=\left\{ \left( a,d\right) \right\} \), \(P_{3}=\left\{ \left( a,c\right) \right\} \), \(P_{4}=\left\{ \left( b,a\right) \right\} \), and \(P_{5}=\left\{ \left( c,b\right) \right\} \). These semiorders induce choices \(C\left( a,b,c,d\right) =a\), \(C\left( a,b,c\right) =b\), and \(C\left( b,c\right) =c\), violating OD.
Proposition 7
\(a\succ b\succ c\succ d\) and \(\mathcal {N=}\left\{ \left( b,\left\{ a,b\right\} \right) ,\left( b,\left\{ b\right\} \right) ,\left( c,\left\{ a,c\right\} \right) ,\left( c,\left\{ c\right\} \right) \right\} \) induce a choice function \(C^{\succ ,\mathcal {N}}\) that violates SH with respect to \(\left\{ a,b\right\} \) and \(\left\{ a,c\right\} \).
Proposition 9
\(C_{5}\) satisfies NBCC and violates OD.
Proposition 10
The following choice function satisfies OD but violates LCAW with respect to the set \(\left\{ a,b\right\} \): \(C\left( a,b,c,d\right) =C\left( a,c\right) =c\), \(C\left( a,b,c\right) =C\left( a,b\right) =C\left( b,c\right) =b\), \(C\left( a,c,d\right) =C\left( b,c,d\right) =C\left( a,d\right) =C\left( b,d\right) =C\left( c,d\right) =d\), and \(C\left( a,b,d\right) =a\).
The following filter \(\Gamma \) satisfies In and Ir: \(\Gamma \left( S\right) =S\) for all \(S\in \mathbb {X}\) with the exceptions \(\Gamma \left( a,b,c,d\right) =\Gamma \left( a,c,d\right) =\Gamma \left( b,c,d\right) =\left\{ c,d\right\} \), \(\Gamma \left( a,b,c\right) =\left\{ b,c\right\} \). Together with \(a\succ b\succ c\succ d\) the filter induces a choice function that violates OD with respect to \(\left\{ a,b,c,d\right\} ,\left\{ a,b,c\right\} \), and \(\left\{ a,b\right\} \).
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Tutić , A. Revealed norm obedience. Soc Choice Welf 44, 301–318 (2015). https://doi.org/10.1007/s00355-014-0830-y
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DOI: https://doi.org/10.1007/s00355-014-0830-y